Chelsea were awarded a penalty after Ryan Gravenberch brought down Moises Caicedo inside the Liverpool area, presenting the visitors with an opportunity to take the lead shortly before half-time.
Henderson initially collected the ball and appeared ready to take responsibility for the spot-kick. However, the midfielder was met with a loud chorus of boos from Liverpool supporters before handing the ball to Rogers, who converted to put Chelsea ahead.
The incident became one of the night's talking points, particularly given Henderson's history at Liverpool and his return to the club's home ground in Chelsea colours.
The former Liverpool captain had also been booed during other passages of play, underlining the reception awaiting him after his departure from Anfield.
Sky Sports highlighted the penalty incident, drawing attention to the sequence in which Henderson picked up the ball before eventually passing it to Rogers. Henderson has not publicly explained his decision to allow his team-mate to take the penalty.
Rogers made no mistake from the spot, sending goalkeeper Giorgi Mamardashvili the wrong way to give Chelsea the lead in the 40th minute.
The advantage was short-lived, however. Liverpool were awarded a penalty of their own after Wesley Fofana handled the ball in the area, and Cody Gakpo converted three minutes later to restore parity.
The teams went into the interval level at 1-1, and neither could find a decisive goal during the second half. Chelsea had opportunities to regain the lead, while goalkeeper Mike Penders made important saves to keep Liverpool at bay.
With the tie unresolved after 90 minutes, the fourth-round contest went to a penalty shootout to determine which side would progress to the quarter-finals.
Henderson again became involved from the spot, this time taking responsibility himself. After Liverpool's Kostas Tsimikas missed an opportunity to secure victory, Henderson converted his penalty to keep Chelsea in the contest.
His successful effort briefly restored the balance in a tense shootout, but Liverpool eventually prevailed 6-5.
Ryan Gravenberch missed for the home side, while Chelsea debutant Emmanuel Emegha also failed to convert. Joe Gomez then scored for Liverpool before Fofana sent his attempt over the crossbar, confirming Chelsea's elimination.
The result secured Liverpool's place in the Carabao Cup quarter-finals, while Chelsea were left to reflect on a painful defeat after remaining competitive throughout the match.
For Henderson, the evening brought a mixture of hostility and personal involvement. He had been subjected to boos from supporters of the club he once captained, handed over a penalty during normal time and then scored from the spot in the shootout.
Liverpool ultimately had the final say, but Henderson's return to Anfield provided one of the night's most striking individual moments in a contest decided by the narrowest of margins.